This study is designed to determine if stem and progenitor cells from umbilical cord blood units (UCBUs) are a clinically acceptable alternative to those from marrow or peripheral blood for unrelated-donor allogeneic transplantation. Questions to be answered about UCBU transplantation include: 1) can children unrelated to the neonate-donors be transplanted with a graft failure rate no greater than 5-10 percent? 2) can adults and larger children (>40 Kg) also be transplanted successfully (all successful matched sib and matched unrelated-donor cord blood transplants to date have been for patients weighing less than 40 Kg)? 3) is graft-vs-host disease (GvHD) usually mild and easily manageable? 4) how much and what kind of HLA disparity can be tolerated? 5) is there enough graft-vs-leukemia (GvL) effect to prevent relapse? 6) what is the course of immune reconstitution? To answer these and other questions about UCBU transplantation will require establishing: a) two to four collection and storage centers (Cord Blood Banks - CBBs) to provide the necessary resource by collecting, processing, cryopreserving and distributing human umbilical cord blood for transplantation to unrelated recipients; b) six to eight Cord Blood Transplant Centers (CBTCs) to follow common protocols for the transplantation of cord blood stem and progenitor cells; and c) a Medical Coordinating Center (MCC) to manage donor searches, facilitate the provision of cord blood units for transplants and collect and analyze data from these transplants. The purpose of this program is to accelerate and oversee clinical research in the rapidly emerging use of umbilical cord blood in place of marrow for transplantation.